Several readers—thanks to James, Steve E, and Ted Poppke—found the story “When U.S. air force discovered the flaw of averages“, which is an excerpt from the Todd Ross book The End of Average.
Story is this. Mid-last-century, an Air Force—ooh rah!—lieutenant was dispatched to measure body shapes of pilots. Everything “including thumb length, crotch height, and the distance from a pilot’s eye to his ear” was put to the tape, and, as statistical practice goes, averages were taken.
The lieutenant (Daniels) was experienced in his duty and had measured college students before.
Even more surprising, when Daniels averaged all his data, the average hand did not resemble any individual’s measurements. There was no such thing as an average hand size…
Using the size data he had gathered from 4,063 pilots, Daniels calculated the average of the 10 physical dimensions believed to be most relevant for design, including height, chest circumference and sleeve length. These formed the dimensions of the “average pilot,” which Daniels generously defined as someone whose measurements were within the middle 30 per cent of the range of values for each dimension. So, for example, even though the precise average height from the data was five foot nine, he defined the height of the “average pilot” as ranging from five-seven to five-11.
Lo, none of the four thousand pilots fit the average on all the dimensions.
Daniels discovered that if you picked out just three of the ten dimensions of size — say, neck circumference, thigh circumference and wrist circumference — less than 3.5 per cent of pilots would be average sized on all three dimensions. Daniels’s findings were clear and incontrovertible. There was no such thing as an average pilot. If you’ve designed a cockpit to fit the average pilot, you’ve actually designed it to fit no one.
There’s more to the story about a supposed ideal female that you’ll want to scan.
Why isn’t there an average pilot? Suppose height was the sole engineering concern. Then 30 percent of the men measured would be average, because the middle 30 percent would be the middle 30 percent, n’est-ce pas?
Next suppose right arm length and height were of concern. Now if whatever caused a man’s height also, and in step (speaking loosely), caused his arm length, then every man’s height would track his arm length, and again 30 percent of the men measured would be average.
But since there are many causes of height (genetics plus environment), and many different, plus some similar, causes of arm length, then the two measures won’t track exactly. Because the causes differ, and although there will be some overlap, the same 30 percent of men in the middle of height won’t be the precise set of men in the 30 percent middle of arm length.
As the number of dimensions increases, the causes become more diverse and the men more unlike one another (across all the dimensions). Simple as that.
It would be a tremendous but common mistake to speak of “normal distributions”, “correlations”, and so forth to say why men don’t match the mean. None of these statistical abstractions are real. The measurements are real, the causes of the things measured are real. But “normal distributions” and “correlations” are not real.
Thus it is a mistake to say “height and arm length are normally distributed” and another mistake to say “height and arm length have such-and-such a correlation.” Nothing in the world is “normally distributed” because normal distributions don’t exist.
Our uncertainty in the unknown heights and arm lengths of future pilots (and not the ones already measured) might, at crude approximation only, be represented using a normal distribution, and the uncertainty we have in the relationship between these two measures might crudely be summarized using a correlation, but that’s all statistics can do. It remains mute on what the causes of the measurements are.
Now no one man may fit the fuzzy average across several dimensions, but it could still be that there may be groups who may who cluster around other measures beside the mean. That is, no man might be within plus-or-minus 30 percent of the average-of-all-dimensions, but some men might be within some plus-or-minus 30 percent of some-function-of-all-dimensions (which isn’t the average). What might these functions be?
Well, anything. Could be height multiplied by arm length divided by inter-eye distance all added to elbow thickness is one function which identifies a large number of similar men, who we could then say fit a type.
Since the number of possible functions increases with the number of dimensions measured, it becomes quite a chore to find representative ones. There are many functions in math! So if we want to do this algorithmically, we have to limit the functions to certain kinds (addition and multiplication, say). Or if we want to do a real bangup job, we could hunt for the various causes of the dimensions.